So, 네 is like an abbreviation of 너의. 의 is used to show property kind of like the english 's. So that is fine to say, but as you mentioned, that is used in casual language or 반말. 당신 is a polite way to say you, so you could say 당신의 생일, but it might sound a bit stiff. A lot of times the word "you" is omitted since you can usually tell by context who's birthday you are asking. For example, 생일이 몇월 몇일이에요? If you are talking directly to one person, what are the odds you are asking your own birthday? So they often omit words like "you" or "I". Hope that helps.